Dec 2020

##### Golden Ratio, or Golden Mean

It is suggested that the great artistic Masters of the past, in particular, had a variety of ‘Rules of Composition’ to help them maximise the aesthetic quality of their art.

Aristides, in her book ‘Classical Painting Atelier’, begins with an introduction saying that a lot of the skills and the value placed on composition were laid aside over the last 150 years. She has an interesting quote from Jacques Villion ‘In the artistic chaos of these last years, when the absolute liberation of the individual instinct has brought it to the point of frenzy, an attempt to identify the harmonic disciplines that have, secretly, in every period, served as foundations for painting may well seem folly. Yet the framework of art is its most secret and its deepest poetry’.

In ‘Classical Drawing Atelier, Aristides suggests that the Greeks, and perhaps even the Egyptians, some 1000 years earlier, were aware of a design aesthetic that has become known as the ‘Golden Ratio’. She says Master artists of the past often took their inspiration from designs and repeating patterns they saw in nature. When mathematicians analysed some of these repeating patterns they found they had an order which could be expressed as a ratio, ‘a proportional ratio that is responsible for the order of design found in much of nature and man’, and ‘for the beauty within visual design’.

The Golden Ratio establishes a relationship so that ‘the proportion of the larger part to the smaller part is the same as the whole to the larger part’. This is explained visually in the diagram below ‘using a design that occurs frequently in the natural world – the logarithmic spiral.’ From this also comes what most people know of as the ‘Fibonacci sequence’.

Starting from the left with one square, it is shown how this sequence builds, and highlights the relationships from that initial square to the squares thereafter, building from that one starting point, to the next in a spiral direction.

The numerical sequence, as shown in the above illustrations begins 1,1,2,3,5,8… and continues where you add the two previous numbers together to get the next number in the sequence. If you divide a number by its preceding number, the ratio is approximately 1.618, and this doesn’t alter, even as the numbers get larger. This is known as the Golden Ratio number (or Phi).

If you then draw a spiral through the corners of each square, it creates a ‘logarithmic’ spiral…

Many examples of how this ordered idealized spiral pattern occurs in nature can be found.

So how then does knowing about this Golden ratio of 1:1.618 help when composing art?

It is suggested that putting key features of the artwork at any of the points that correspond with this ratio improves the aesthetic quality of the art. It was funny, for all the hype about ‘Great masters used this method’, it was relatively difficult to track down examples which showed this specifically. These two examples show the artwork overlaid with the logarithmic spiral rather than just a set of rectangles that might have done just as well. Putting in the spiral arguably complicates the issue of the 1:1.618 ratio.

I also don’t really understand why the rectangle is only overlaid over part of the image and not the whole. Surely then it must be possible to select lots of examples in many paintings that use the Golden ratio in one part or another if you can be this selective, and then the question arises as to whether the artist was actually using that at all in planning out their artwork? I understood the device was for composing the whole of the artwork.

Similarly this example only places the spiral over part of ‘The Great Wave off Kanagawa, by Hokusai. (1829-1833).

Aristides uses Leonardo da Vinci’s ‘The Annunciation’ for her example. Rectangle A and C, represented by the character ‘Phi’ are both golden ratio rectangles. Although B is a square, A+B is a golden ratio, as is B+C, so in all there are 4 golden ratio rectangles in this work.

Taking this even further, and using the centre of the logarithmic spiral for each of A+B and B+C she shows that ‘the long diagonals in the side rectangles…delineates key areas of the composition’. This aims to show how Da Vinci fully exploited the aesthetics of the Golden ratio.

__Rule of thirds__

__Rule of thirds__

Harmonic ratios are another aspect of composition based on findings by Pythagoras, which originated through his observations about sounds and in particular, when notes produced harmonic, or pleasing, effects. He devised mathematic principles from these investigations which artists later adopted as ratios to divide up their canvases and aid their compositions to create visual harmony.

One particular compositional method is ‘The Rule of thirds’. In this image from Aristides (Painting) you can see points at which the blue and green diagonal lines cross at a third of the way across the page, vertically and horizontally. These points are known as ‘the eyes of the rectangle’ and mark the points corresponding to the ‘Rule of thirds’. Like the Golden ratio, to optimise aesthetic value and harmony in your artwork, it is advocated that when planning the composition of your work, that you position important aspects of your work at these points.

Whilst working on other research for this Unit about Subjective landscapes, I came across this painting ‘Christ with Angels’ by Moreau, and could see readily that Moreau has taken the rule of thirds into account in designing his composition. Nearly all the main figures faces lie along one vertical third line and the main angels face lies at an intersection with a horizontal third. The horizon lies along the other third line.

Aristides uses a painting by Rubens to demonstrate the use of the ‘Rule of thirds’, but she takes into account the diagonal lines that are used to construct these points too, as design devices, and in that the complexity of this composition can be more fully appreciated.

Aristides goes on to build more lines to demonstrate the harmonic ratios which she calls the ‘Armature of the rectangle’. Any of these can also be used as compositional devices and refers to it as ‘the musical scale of composition’. She takes the reader step-by-step through how to prepare this framework, and then demonstrates in the finished diagram how it relates to the sin waves of musical notes. This framework also includes the ‘Rule of Thirds’ lines and points.

She adds that this framework can ‘be used to create an endless variety of compositions, from simple to complex, all the while retaining a system of order found within nature.’

She demonstrates how Diego Valazquez might have used some of these devices in his composition for Las Meninas, 1656. Many of the lines cross key aspects of the characters and the setting in this artwork. She makes a compelling argument for this, and it appears there is so much ‘order’ and correlation with the compositional devices that they surely cannot have been created merely by chance.

In summary, the ‘Golden Ratio’ (or ‘Mean’) and the ‘Rule of Thirds’ are two of a number of compositional devices that can be employed when experimenting with the composition of a painting. It is good to be aware of them, and interesting to find out more about them and about some of the other devices that Aristides makes reference to. The rule of thirds is much easier to understand and employ and indeed when taking photographs I always have a third-grid switched on my camera so that I can experiment with this in my compositions. For many of my landscape paintings in this unit too, I have considered ‘the Rule of Thirds’ as an aspect of my composition. It is quite automatic for me. However, it is clear that there is more to this device than merely positioning points of interest at particular intersections or just along the horizontal and vertical third lines themselves. The significance of diagonal construction lines and compositional triangles, squares and circles for example, within artwork are all further considerations for me for my future work.

__References__

**ARISTIDES, **Juliette, (2008), *‘Classical Painting Atelier – A Contemporary Guide to Traditional Studio Practice’, *Watson-Guptill Publications, New York

**DA VINCI, **Leonardo, (1472-1475), ‘*The Annunciation’ *Classical Drawing Atelier – A *Contemporary Guide to Traditional Studio Practice’, 2006, *Watson-Guptill Publications, New York

**DA VINCI, **Leonardo, (1490), *‘Vetruvian Man’, *‘Learn the golden Ratio for Your Artworks on canvas’UCS Upper Canada Stretchers, 2017 [online]. Available at: https://www.ucsart.com/learn/blog/learn-the-golden-ratio-for-your-artworks-on-canvas [Accessed 5.11.2020]

**DOROSHENKO, **V, (2016), *Mathematical Relations for Harmonization with Technical and Decorative Casting Nature’, *ResearchGate [online]. Available at: https://www.researchgate.net/figure/Examples-of-the-presence-of-Archimedes-spiral-and-Fibonacci-numbers-in-nature_fig1_313827125 [Accessed 5.1..2020]

**DR MATH, **(2020), *‘Golden Ratio, Fibonacci Sequence’. *The Math Forum [online]. Available at: http://mathforum.org/dr.math/faq/faq.golden.ratio.html [Accessed 4.11.2020]

**FIBONACCI SPIRAL **(2020), *Fibonacci Numbers’, *Uncredited, unnamed website. [online] Available at: http://www2.nau.edu/lrm22/lessons/fibonacci/fibonacci.html [Accessed 4.1..2020]

**HOKUSAI, **K, (1829-1833) *‘The Great Wave off Kanagawa’*, 11:11 – Is it happening to You? Marie Jones and Larry Flaxman, Time is Art – A Documentary Series, via New Dawn Magazine 2020 [online] Available at: https://thesyncmovie.com/2017/03/02/1111-is-it-happening-to-you/ [Accessed 5.11/2020]

**MOREAU, **Gustav, (1685) *‘Christ with Angels’, *Bridgeman Education. [online]. Available at: https://www-bridgemaneducation-com [Accessed 17.11.2020]

**RUBENS, **Peter Paul, (1613), *Venus, Cupid, Bacchus and Ceres’, *‘Classical Painting Atelier – A Contemporary Guide to Traditional Studio Practice’, 2008, Watson-Guptill Publications, New York

**VALAZQUEZ, **Diego, (1656), ‘*Las Meninas, *‘Classical Painting Atelier – A Contemporary Guide to Traditional Studio Practice’, 2008, Watson-Guptill Publications, New York